Method and control unit for controlling an overdetermined system, system and aircraft

ABSTRACT

A method for controlling an overdetermined system with multiple power-restricted actuators that perform a primary task and non-primary tasks, including: a) determining a pseudo-control command based on a physical model of the system, which pseudo-control command represents the torques and a total thrust force acting on the system, b) determining a control matrix, c) dissociating the control matrix into sub control matrices, wherein the sub control matrices and the corresponding sub pseudo-control commands correspond to the primary task for i=1 and for i&gt;1 correspond to the non-primary task(s) and a priority of the non-primary tasks decreases with increasing index i, d) determining actuator control commands for solving the primary task, e) projecting the non-primary tasks into the null space of the primary task, and into respective null spaces of all of the non-primary tasks of higher priority, if present, and f) providing the actuator control commands from d) and e) at the actuators.

INCORPORATION BY REFERENCE

The following documents are incorporated herein by reference as if fullyset forth: German Patent Application No. 10 2021 111 104.8, filed Apr.29, 2021.

TECHNICAL FIELD

The invention relates to a method for controlling an overdeterminedsystem with multiple actuators, for example an aircraft with multiplepropulsion units.

The invention also relates to a control unit for controlling anoverdetermined system with multiple actuators, for example an aircraftwith multiple propulsion units.

The invention further relates to an overdetermined system with multipleactuators.

Finally, the invention also relates to an aircraft with multiplepropulsion units and optionally further actuators, such as movable flapsor winches, which together form an overdetermined system.

BACKGROUND

Airplanes, aerial vehicles or aircraft with overdetermined actuation(“over-actuation”, i.e. the airplane has a larger number of actuatorsthan is necessary for the execution of so-called primary tasks in itscontrol volume) offer the possibility of realizing a primary task withmore than one solution. This intuitively implies that, depending on theairplane type, the actuator type and the tasks defined in the controlvolume, there may be additional room to perform one or more other(non-primary) tasks.

Depending on the airplane maneuver, environmental conditions, failurescenarios or other effects, it may happen that the airplane is operatedclose to the limits of its control volume. This means that the physicalforces and moments (torques) that can be provided by the actuators andcontrol surfaces (flaps, wings, etc.) are possibly not sufficient tocontrol all of the control axes of the airplane at all points in time—inthe case of the “VoloCity” multicopter produced by the applicant theseare the rolling axis (longitudinal axis), the pitching axis (transverseaxis), the yawing axis (vertical axis) and the (total) thrust. Anotherairplane type may, depending on the arrangement of the actuators, havedifferent control axes, in particular two additional horizontal thrustaxes. In such situations, the safety and controllability of the airplanecan be ensured by one or more control axes being prioritized over theothers, for example the rolling and pitching axes over the yawing axis.It is in this case important to ensure that a control axis with lowpriority does not adversely affect the control axis (axes) with highpriority (for example if yawing has a lower priority than rolling andpitching, the yawing control must not lead to an adverse effect onrolling and pitching).

This isolation of different priorities in the control is not a trivialtask. Generally, combining different tasks with a respective priority insuch a way that low-priority tasks do not affect high-priority tasks isa complex challenge. The way secondary or third-level (tertiary orgenerally non-primary) tasks are added to a control and allocationproblem (assignment problem) can substantially affect the availableperformance for solving a primary task and the stability of thatsolution.

In European patent application 19 212 935.1, the applicant proposes amethod for operating an eVTOL aerial vehicle with 18 rotors, in whichthe primary control assignment finds a solution with so-called “L-2”minimization for the provision of the required thrusts and torques,while imposing a secondary and lower priority (not primary) compulsionto reduce the maximum LTU (lift thrust unit) power. This is carried outby changing the gain parameters of the assignment problem as a functionof the deviation of an LTU value (the revolution rate—RPM) from thecorresponding average value of the total LTUs (so-called “L-inf”minimization). Two tasks are carried out in one step, which makes itimpossible to assign a different priority or criticality to thedifferent goals. In addition, a failure or erroneous behavior whenachieving a goal with lower criticality (for example a secondary goal)can negatively affect solving a task with higher criticality (forexample a primary goal).

SUMMARY

The invention is based on the object of remedying the situation here andspecifying a method, a control unit, a system and an aerial vehicle oran aircraft of the respective type mentioned, with which, in addition tothe solution of at least one primary task, the solution of at least onenon-primary task is possible without adversely affecting the primarytask or its solution. The same is generally intended to apply torelatively higher-priority tasks over relatively lower-priority tasks.

This object is achieved by a method with one of more of the featuresdisclosed herein, by a control unit with one of more of the featuresdisclosed herein, by a system with one of more of the features disclosedherein, and by an aircraft with one of more of the features disclosedherein.

Preferred further developments are defined below and in the claims.

A method according to the invention for controlling an overdeterminedsystem with multiple power-restricted actuators (number k), inparticular an aircraft with multiple propulsion units, wherein theactuators perform at least one primary task and a number of non-primarytasks, includes:

-   -   a) determining a pseudo-control command u_(p)∈        ^(p′) based on a physical model of the system, which        pseudo-control command represents in particular the torques (L,        M, N) and a total thrust force (F) acting on the system,    -   b) determining a control matrix D, D∈        ^(p′×k), according to u_(p)=Du, wherein u denotes actuator        control commands,    -   c) dissociating the control matrix D into sub control matrices        D_(i) according to

${D = \begin{bmatrix}D_{1} \\ \vdots \\D_{q}\end{bmatrix}},$

q≤p′, so that

${u_{p} = {{\begin{bmatrix}u_{p,1} \\ \vdots \\u_{p,q}\end{bmatrix}{and}u_{p,i}} = {D_{i}u}}},$

wherein the sub control matrices D_(i) and the corresponding subpseudo-control commands u_(p,i) for i=1 correspond to the primary taskand for i>1 correspond to the at least one non-primary task and whereina priority of the non-primary tasks decreases with increasing index i,

-   -   d) determining the actuator control commands for solving the        primary task according to u₁=D₁ ^(#)u_(p,1), where # denotes a        matrix inversion and u₁∈        ^(k) comprises those actuator control commands which solve the        primary task, with u₁∈U:={u∈        ^(k)|u^(min)≤u≤u^(max)},    -   e) projecting the non-primary tasks, i>1, into the null space of        the primary task, i=1, and into the respective null space of all        of the non-primary tasks of higher priority, if present, so that        D_(i)u_(j)=0, with u_(j)∈        ^(k), j>i, comprises actuator control commands for performing a        non-primary task, and    -   f) providing the actuator control commands from d) and e) at the        actuators.

The term “restricted” or “power-restricted” means in this context thatthe actuators have physical power limits, that is to say cannot performevery conceivable technical task (which applies in principle to anymachine).

It correspondingly applies for the actuator control commands u, thatu∈U:={u∈

^(k)|u^(min)≤u≤u^(max)}⊂

^(k) denotes a specific range of values of restricted actuator controlcommands.

A control unit according to the invention for controlling anoverdetermined system with multiple power-restricted actuators (numberk), in particular an aircraft with multiple propulsion units, whereinthe actuators perform at least one primary task and at least onenon-primary task, is designed and set up, in particular softwaretechnically, for

-   -   a) determining a pseudo-control command u_(p)∈        ^(p′) based on a physical model of the system, which        pseudo-control command represents in particular the torques (L,        M, N) and a total thrust force (F) acting on the system,    -   b) determining a control matrix D, D∈        ^(p′×k), according to u_(p)=Du, wherein u denotes actuator        control commands,    -   c) dissociating the control matrix into sub control matrices

${D = \begin{bmatrix}D_{1} \\ \vdots \\D_{q}\end{bmatrix}},$

q≤p′, so that

${u_{p} = {{\begin{bmatrix}u_{p,1} \\ \vdots \\u_{p,q}\end{bmatrix}{and}u_{p,i}} = {D_{i}u}}},$

wherein the sub control matrices D_(i) and the corresponding subpseudo-control commands u_(p,i) for i=1 correspond to the primary taskand for i>1 correspond to the at least one non-primary task and whereina priority of the non-primary tasks decreases with increasing index i,

-   -   d) determining the actuator control commands for solving the        primary task according to u₁=D₁ ^(#)u_(p,1), where # denotes a        matrix inversion and u₁∈        ^(k) comprises those actuator control commands which solve the        primary task, with u₁∈U:={u∈        ^(k)|u^(min)≤u≤u^(max)},    -   e) projecting the non-primary tasks, i>1, into the null space of        the primary task, i=1, and into the respective null space of all        of the non-primary tasks of higher priority, if present, so that        D_(i)u_(j)=0, if u_(j)∈        ^(k), j>i, comprises actuator control commands for performing a        non-primary task, and    -   f) providing the actuator control commands from d) and e) at the        actuators.

The control unit may be further designed and set up to carry out furtherdevelopments described below of the method according to the invention.

Furthermore, the control unit may be actively connected to devices formeasuring and/or determining parameters and states of the system and/orthe actuators which are required for the determination of thepseudo-control command according to step a), in particular by a mainflight control unit of an aircraft.

An overdetermined system according to the invention comprises multipleactuators actively connected to a control unit according to theinvention.

An aircraft according to the invention comprises multiple propulsionunits and optionally further actuators, such as movable flaps orwinches, which together form an overdetermined system according to theinvention, in particular in the form of a multicopter with a number ofpreferably electrically driven rotor units.

“Actuators” are understood in the present case to mean in particular,but not exclusively, propulsion units, such as rotor units or the like.In addition, depending on the design of the system (the aerial vehicle),there are other means, such as flaps, winches, etc. or generally meansof operation, which the system needs to perform certain physical tasks.

A method for assigning control priorities to overdetermined systems, andspecifically airplanes, in which low-priority tasks are projected intothe null space of higher-priority tasks is proposed. This ensures thatthe low-priority tasks do not adversely affect the high-priority tasks.

A method for prioritizing control axes of a system (airplane) withoverdetermined actuators by using its null space is proposed here inparticular. With the method presented here, in the case of an airplanethe control of the rolling axis can be prioritized over the control ofthe pitching axis, or the control of the rolling and pitching axes overthe control of the yawing axis, or the control of the rolling andpitching axes over the control of the yawing axis over the altitudecontrol, and so on. The combination of the priorities may be chosen asdesired, and an underlying algorithm, which may preferably beimplemented in software form in one configuration of the control unitaccording to the invention, can be applied to all such combinations.

It can be seen as a preferred feature that, on account of the use of thenull space of the control matrix, a goal/task with lower priority cannotaffect the task(s) with higher priority. For example, a tertiary taskcannot affect the secondary and primary tasks, and a secondary taskcannot affect the primary task.

The presented method can also be used for quaternary, quinary or othertasks if there is still appropriate room in the null space. If there isno more room in the null space, lower-priority tasks are advantageouslysimply ignored due to the definition of the null space.

In the present case, “null space” is to be understood as the core of a(linear) mathematical representation. The core of a linearrepresentation f: V→W between vector spaces V, and W consists of thosevectors in V which are mapped onto the zero vector in W; it is thereforethe solution set of the homogeneous linear equation f(x)=0 and istherefore also referred to as a null space.

Looking at the example from EP 19 212 935.1, the primary task may be tofind the solution of the control assignment to generate the required(total) thrust and the required torques with “L-2 minimization”.Accordingly, a secondary, i.e. non-primary, task can be the reduction ofthe maximum LTU power with “L-inf minimization”. If this secondary taskis projected into the null space of the primary task according to thepresent invention, it is ensured that it does not affect the solution ofthe primary task. This is desirable for safety reasons, as the primarytask is essential, but the secondary task is only preferable. Inaddition, another subordinate (for example tertiary task) can beassigned, for example reducing the power of a particular (or multiple)LTU(s) to for example 75% because the propulsion unit in question hasoverheated or was classified as faulty. As part of a correspondingfurther development of the invention, the tertiary task is projectedinto the null space of the primary and secondary tasks, so that it doesnot adversely affect the higher-prioritized tasks.

In the example above, the secondary and tertiary tasks can swap placesdepending on the requirement and prioritization. Within the scope of thepresent application, the priorities of the individual (control) tasksresult from the respective specific application (more details on thisfurther below).

It should be emphasized that the idea proposed here provides anextremely advantageous separation of tasks according to their respectivecriticality, which can be used for assigning different design assurancelevels (DAL) of functions with different criticalities and goals. Thishas a noticeable influence on the reduction of the development costs ofsafety-critical devices, such as specifically aerial vehicles.

Although reference has been repeatedly made here and below to aerialvehicles, the described method is also generally suitable forcontrolling overdetermined actuator systems of any kind.

At this point, first the mathematical-physical background of theinvention—in particular with regard to aerial vehicles—will be brieflydiscussed in more detail in order to facilitate understanding.

The equations of motion of systems which can be derived with theNewton-Euler principle or the Lagrange method can be represented asfollows:

M(x){umlaut over (x)}+c(x,{dot over (x)})+g(x)+G(x)u _(p)=ƒ_(ext),  Equation 1

wherein x∈

^(m) denotes the m-dimensional configuration vector of the system, forexample positions and rotations in 3D, M(x)∈

^(m×m) denotes the state-dependent generalized inertial torque, (c,ċ)∈

^(m) denotes the state-dependent Coriolis forces, g(x)∈

^(m) denotes the gravitational forces, and ƒ_(ext)∈

^(m) are the external forces and torques, for example due toaerodynamics, contact, etc. The required physical control commands (orpseudo-control commands) for the system are designated as u_(p)∈

^(p′), which are for example calculated with a feedback control law andused to control the system. These pseudo-control commands are thebody-fixed forces and torques acting on the system through differentactuators, and they each enter the system dynamics given in equation 1with a rule input matrix G(x)∈

^(m×p′). These matrices contain in particular the information about anover- or under-actuation.

For the calculation of u_(p), control methods (or laws) are used (forexample direct dependencies or feedback control laws, etc.). Theconnection between these calculated control commands and the actualactuator control commands u_(p)u∈

^(k) is made by means of an assignment or allocation problem, whichcontains in particular the geometric knowledge about the positioning ofthe actuators in the system and other actuator-relevant configurationsand properties. The following applies:

u_(p)=Du,   Equation 2

with u∈U:={u∈

^(k)|u^(min)≤u≤u^(max)}⊂

^(k), wherein D∈

^(p′×k) defines the so-called control effectiveness matrix (in short:control matrix). According to the use of control methods—as mentionedabove—the pseudo-control commands u_(p)∈

^(p′) are first calculated. However, these must be distributed to thephysical actuators in the form of the actual control commands u∈

^(k) (actuator control commands), which is commonly known as the controlmapping problem or allocation problem. Therefore, a type of inversematrix calculation (identified below by the superscript “#”) is requiredto calculate u from u_(p). This is represented by

u=D ^(#)(W,u ^(min) ,u ^(max))u _(p),   Equation 3

wherein this inversion is usually carried out taking into account aweight or weighting matrix W_(i)∈

^(k×k) and the physical limits of each actuator, for example u^(min)∈

^(k) and u^(max)∈

^(k), wherein ∀i=1, . . . , k:u_(k) ^(min)≤u_(i)≤u_(k) ^(max).

A=D^(#)(W,u^(min),u^(max))∈

^(k×p′) is referred to as the “control mapping matrix” (or theallocation matrix).

Note: If M∈

^(k×p) is a matrix with rank(M)=p, then M^(#) denotes the right inverse,so that M M^(#)=I. It is the standard matrix inverse M^(#)=M⁻¹ for k=p′and—ignoring the weighting matrix W and the limits u^(min), u^(max)—thepseudo-inverse M^(#)=M^(T)(M M^(T))⁻¹ for k>p′.

For an aerial vehicle, such as the 18-rotor Volocopter® (model “VC200”,“VoloCity” or “VoloDrone”) from the applicant's company, u_(p)∈

⁴ represents a vector which contains the required (control) thrust andthe three-dimensional (actuation) torques which act on the aerialvehicle with regard to its center of gravity. Furthermore, u∈

¹⁸ is a vector which contains the eighteen (18) desired rotor controlcommands.

The distributed actuator control commands u were calculated above usingthe mapping matrix A∈

^(k×p′), which is a result of a matrix inversion problem. For systemswith a redundant number of actuators (i.e. k>p′), there may be more thanone solution to this inversion problem if the control matrix has enoughlinearly dependent columns (this is known to those skilled in the artwho are familiar with linear algebra). In this application, such systemsare referred to as “overdetermined systems”.

The presence of more than one solution to the inversion problem createsthe possibility of a “null space” different from zero which can be usedfor other (secondary or tertiary or generally lower-prioritized) taskswithout affecting overarching goals (tasks or their solutions).

Definition of the Control Axes

The (control) matrix D establishes a relationship between the physicalcontrol axes u_(p) (for example torques and thrust) and the actualactuator commands u on the basis of the relationship u_(p)=Du. To allowprioritization of the control axes, the following division is proposed:

$\begin{matrix}{{u_{p} = \begin{bmatrix}u_{p,1} \\u_{p,2} \\ \vdots \\u_{p,q}\end{bmatrix}},} & {{D = \begin{bmatrix}D_{1} \\D_{2} \\ \vdots \\D_{q}\end{bmatrix}},} & {{q \leq p},}\end{matrix}$

where: u_(p,i)=D_(i)u. Here, u_(p,i) denotes a subset of the controlaxis concerned. It may be a single control axis, if dim(u_(p,i))=1, ormultiple control axes, if 1<dim(u_(p,i))<p.

For example, u_(p,1) may be the rolling and pitching moment(dim(u_(p,1))=2), u_(p,2) may be the yawning moment (dim(u_(p,2))=1),and u_(p,3) may be the thrust (dim(u_(p,3))=1).

Null Space

The null space, also known as the core, of a matrix D_(i)∈

^(p′×k) is the set of all vectors v, so that D_(i)v=0. Therefore, thenull space of the matrix D_(i)v=0 can be defined as:

N(D _(i))={v|D _(i) v=0}.

Calculation of the Null Space

A possible representation of the null space of a matrix D can becalculated according to:

N(D)=I−D ^(T)(D ^(#))^(T)∈

^(k×k),

wherein I∈

^(k×k) is a k×k identity matrix. The superscript character # implies theinversion (standard inversion if the matrix is quadratic, orpseudo-inversion if not), and the superscript character T implies thetransposing operation.

Task Prioritization by Means of Null Space Projection

The control matrix D can be divided into smaller matrices (dissociated),the number of this division, separation or dissociation depending on thenumber of tasks/goals to be accomplished:

D=[D ₁ ^(T) D ₂ ^(T) D ₃ ^(T) . . . D _(q) ^(T)]^(T),

where q≤p is the number of tasks or goals to be performed. The smallerits index, the more important it is for it to be performed, and the taskconcerned (the goal concerned) is prioritized over the othertasks/goals.

For example, the goal with the highest priority is assigned using D₁(corresponding to u_(p,1)). Accordingly, the secondary goal is assignedusing D₂ (corresponding to u_(p,2)), etc.

Example of an Application with Three Tasks/Goals:

Primary Goal

It is the case that

u₁=D₁ ^(#)u_(p,1),

wherein u₁∈

^(k) is the vector which contains the actuator commands (or actuatorcontrol commands) which achieve the primary goal of generating therequired physical control commands u_(p,1). The mapping generatesexecutable commands, with

u ₁ ∈U:={u∈

^(k) |u ^(min) ≤u≤u ^(max)}}.

Secondary Goal

Now considered is u₂∈

^(k), which is calculated to achieve a secondary goal in order togenerate the required physical control commands u_(p,2) withoutdisturbing the primary goal. For this purpose, u₂ is suitably calculatedand projected onto the null space of the control matrix D₁ used for theprimary goal, preferably as follows:

u ₂ ^(org) =D ₂ ^(#)(u _(p,2) −D ₂ u ₁),

N ₂ =I−D ₁ ^(T)(D ₁ ^(#))^(T),

u₂=N₂u₂ ^(org).

A corresponding further development of the method according to theinvention provides that first u₂ ^(org)∈

^(k) is calculated, in order to solve a first, non-primary task,preferably according to u₂ ^(org)=D₂ ¹⁹⁰ (u_(p,2)−D₂u₁), andsubsequently u₂ ^(org) is projected onto the null space N₂(D₁) of thesub control matrix D₁ according to u₂=N₂u₂ ^(org), wherein preferablyN₂=I−D₁ ^(T)(D₁ ¹⁹⁰ )^(T).

In order to ensure the feasibility of the overall command so that u∈U,preferably the following limits must be adhered to in the secondaryassignment step:

u ₂ ^(org) ∈U:={u ₂ ^(org)∈

^(k) |u ^(min) −u ₁ ≤N ₂ u ₂ ^(org) ≤u ^(max) −u ₁}.

A corresponding further development of the method according to theinvention provides that for u₂ ^(org) it is specified that u₂^(org):={u₂ ^(org)∈

^(k)|u^(min)−u₁≤N₂u₂ ^(org)≤u^(max)−u₁}. It follows from this that:u₁+u₂∈U.

The complete actuator command, which ensures that the primary goal isnot affected by the secondary goal (if there is no tertiary goal),preferably looks as follows:

u=u ₁ +u ₂.

A corresponding further development of the method according to theinvention provides that a complete actuator control command u iscalculated as follows: u=u₁+u₂.

Tertiary Goal

Considered is u₃∈

^(k), which is calculated to achieve a tertiary goal in order togenerate the required physical control commands u_(p,3) without at thesame time disturbing either the primary goal or the secondary goal. Thenu₃ is projected onto the null space of the control matrices D₁, D₂ usedfor the primary goal and for the secondary goal, preferably accordingto:

${u_{3}^{org} = {D_{3}^{\#}\left( {u_{p,3} - {D_{3}\left( {u_{1} + u_{2}} \right)}} \right)}},{N_{3} = \left( {I - {\begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{T}\left( \begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{\#} \right)^{T}}} \right)},{u_{3} = {N_{3}{u_{3}^{org}.}}}$

A corresponding further development of the method according to theinvention therefore provides that subsequently u₃ ^(org)∈

^(k) is calculated in order to solve a second, non-primary task,preferably according to u₃ ^(org)=D₃ ^(#)(u_(p,3)−D₃(u₁+u₂)), andsubsequently u₃ ^(org) is projected onto the null space N₃(D₁, D₂) ofthe sub control matrices D₁, D₂ according to u₃=N₃u₃ ^(org), whereinpreferably

$N_{3} = {\left( {I - {\begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{T}\left( \begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{\#} \right)^{T}}} \right).}$

In order to ensure the feasibility of the overall command so that u∈U,the following limits must preferably be adhered to in the tertiaryassignment step:

u ₃ ^(org) ∈U ₃ :={u ₃ ^(org)∈

^(k) |u ^(min) −u ₁ −u ₂ ≤N ₃ u ₃ ^(org) ≤u ^(max) −u ₁ −u ₂}.

A corresponding further development of the method according to theinvention therefore provides that for u₃ ^(org)∈

^(k) it is specified that u₃ ^(org)∈U₃:={u₃ ^(org)∈

^(k)|u^(min)−u₁−u₂≤N₃u₃ ^(org)≤u^(max)−u₁−u₂}. It follows from thisthat: u₁+u₂+u₃∈U.

The complete actuator command, which ensures that the primary goal isnot affected by the secondary goal and the tertiary goal and thesecondary goal for its part is not affected by the tertiary goal, isthen preferably:

u=u ₁ +u ₂ +u ₃.

A corresponding further development of the method according to theinvention therefore provides that subsequently a complete controlcommand u is calculated as follows: u=u₁+u₂+u₃.

This procedure can be generalized:

For the task with the highest priority it always applies that

u ₁ =D ₁ ^(#) u _(p,1).

For the secondary task it applies that

u ₂ ^(org) =D ₂ ¹⁹⁰(u _(p,2) −D ₂ u ₁),

N ₂ =I−D ₁ ^(T)(D ₁ ¹⁹⁰ )^(T),

u₂=N₂u₂ ^(org).

For all other tasks (with i>2) it applies that

u _(i) ^(org) =D _(i) ^(#)(u _(p,i) −D _(i)Σ_(j=1) ^(i−1) u _(j)),

${N_{i} = \left( {I - {\begin{bmatrix}D_{1} \\ \vdots \\D_{i - 1}\end{bmatrix}^{T}\left( \begin{bmatrix}D_{1} \\ \vdots \\D_{i - 1}\end{bmatrix}^{\#} \right)^{T}}} \right)},{u_{i} = {N_{i}{u_{i}^{org}.}}}$

A corresponding further development of the method according to theinvention therefore provides that the null space N_(i)(D), i>1 iscalculated according to

N i ( D ) = ( I - [ D 1 ⋮ D i - 1 ] T ⁢ ( [ D 1 ⋮ D i - 1 ] # ) T ) ∈ k ×k .

Finally, the following is obtained as the complete actuator command

u=Σ_(i=1) ^(n)u_(i).

The limit values are

u _(i) ∈U _(i) :={u _(i)∈

^(k) |u ^(min) −u _(Σ) ≤N _(i) u _(i) ≤u ^(max) −u _(Σ)}

with u_(Σ)=Σ_(j=1) ^(i−1)u_(j).

Specifically, therefore, for solving a non-primary task, i>1, in afurther development of the method according to the invention thefollowing may be calculated:

u _(i) ^(org) =D _(i) ^(#)(u _(p,i) −D _(i)Σ_(j=1) ^(i−1) u _(j))

u_(i)=N_(i)u_(i) ^(org)

u=Σ _(i=1) ^(n)u_(i); with

u _(i) ∈U _(i) :={u _(i)∈

^(k) |u ^(min) −u _(Σ) ≤N _(i) u _(i) ≤u ^(max) −u _(Σ)}

and with

u_(Σ)=Σ_(j=1) ^(i−1)u_(j)

Preferably, the method according to the invention is applied to anaircraft with multiple, k, propulsion units, preferably 18 propulsionunits, k=18, which propulsion units form at least some of the actuatorsof the system.

For example, it may be provided in a further development of the methodaccording to the invention that attitude control concerning rolling (inparticular rolling angles) and pitching (in particular pitching angles)is prioritized over directional control concerning yawing (in particularthe yaw rate) and vertical control concerning the flying altitude (inparticular the climbing and descending rate). This is specificallyexplained in even more detail further below on the basis of FIGS. 1 and2.

In particular, it may be provided here that D₁∈

^(2×k) is chosen such that it represents a mapping of actuator controlcommands u onto the rolling and pitching moment, with u_(p,1)∈

², and that D₂∈

^(2×k) is chosen such that it represents a mapping of actuator controlcommands u onto the thrust and yawing moment, with u_(p,2)∈

² (variant 1).

In another further development of the method, it may be provided thatattitude control concerning rolling and pitching (in particular rollingangles and pitching angles) is prioritized over directional controlconcerning a yawing motion (in particular the yaw rate) and a totalthrust.

In particular, it may be provided here that D₁∈

^(2×k) is chosen such that it represents a mapping of actuator controlcommands u onto the rolling and pitching moment, with u_(p,1)∈

², that D₂∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the yawing moment, with u_(p,2)∈

, and that D₃∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the total thrust, with u_(p,3)∈

(variant 2).

In yet another further development of the method, it may also beprovided that a total thrust is prioritized over attitude controlconcerning rolling and pitching (in particular rolling angles andpitching angles) and over directional control concerning a yawing motion(in particular the yaw rate).

It may be provided for this purpose that D₁∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the total thrust, with, u_(p,i)∈

², that D₂∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the pitching moment, with u_(p,2)∈

, that D₃∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the rolling moment, with u_(p,3)∈

, and that D₄∈

^(1×k) is chosen such that it represents a mapping of actuator controlcommands u onto the yawing moment, with u_(p,4)∈

(variant 3).

The first two variants are aimed at resuming a stable flying position assoon as rolling and pitching angles have returned to a reasonable level.Because this may involve a loss of altitude, the strategy is possiblynot suitable for cases in which the aerial vehicle is close to theground.

By contrast, the third variant is well suited for cases in which acollision is unavoidable, and so instead of prioritizing thrust it canbe attempted to reduce a collision speed in order to minimize damage tothe aerial vehicle. It may also be attempted to reduce the pitchingangle, which increases the crash safety for the pilot and passengers.The same applies to the rolling angle, whereas the yawing angle(directional angle) or yaw rate only has low priority in this case.

BREIF DESCRIPTION OF THE DRAWINGS

Further properties and advantages of the invention result from thefollowing description of the figures.

FIG. 1 shows a system or aerial vehicle according to the invention;

FIG. 2 illustrates the coupling of limit values of the pseudo-controlcommands;

FIG. 3 shows the aerial vehicle from FIG. 1 in an extreme flyingsituation;

FIG. 4 shows a possibility for dividing up the control matrix; and

FIG. 5 shows a possible sequence of the method according to theinvention.

DETAILED DESCRIPTION

FIG. 1 shows a system or aerial vehicle 1 according to the invention inthe form of a multicopter produced by the applicant with specifically 18propulsion units (actuators). In FIG. 1, L, M and N denote the torquesaround the axes x, y and z (roll axis, pitch axis and yaw axis) of theaerial vehicle 1, and F denotes the total thrust. Reference character 2symbolizes the main flight control unit of the aerial vehicle 1, whichpreferably comprises a control unit 2 a according to the invention,which preferably has a control algorithm 2 a′ and as a result is set upfor carrying out the method according to the invention and furtherdevelopments thereof, in particular software technically. In the case ofreference character 2 b, a human pilot is also shown, which is not worthnoting further in the present case. Reference character 3 refers to oneof the 18 (without limitation, identical) propulsion units or actuators,each comprising an (electric) motor 3 a and a rotor 3 b. The aerialvehicle 1 is therefore an eVTOL—an electrically powered aerial vehiclewith the capability of taking off and landing vertically (“verticaltake-off and landing”). Reference character 4 refers to an exemplarysensor unit actively connected to the main flight control unit 2. Inorder to be able to take into account the available aerial vehiclestates in a corresponding further development of the method according tothe invention, a variety of such sensor units 4 may be provided, inparticular inertial measuring units, GNSS, barometers, vibration sensorson the actuators, temperature sensors on the actuators, and the like.

The invention is in principle not limited in its application to aerialvehicles 1 as overdetermined systems.

FIG. 2 illustrates the case explained above, according to which, in thecase of an aerial vehicle, in particular according to FIG. 1, undercertain circumstances rolling angle control and pitching angle controlare prioritized over yawing angles or yaw rate and thrust. According toFIG. 2, this is so because the limit values of the pseudo-controlcommands are coupled such that a great thrust requirement reduces thecontrol range (the control volume, that is to say the shaded area) withrespect to the rolling and pitching moments (abscissa).

For a regular multicopter, the attitude of the aerial vehicle alsodetermines the alignment of the thrust vector. Therefore, the thrustrequirement for compensating the weight of the aerial vehicle increasesgreatly if the attitude angle assumes large values, cf. the illustrationin FIG. 3. If the thrust requirement exceeds the permissible limitvalues, the large attitude angle inevitably has the effect that theaerial vehicle 1 loses altitude. This loss of altitude can only becorrected again when the attitude angle has been reduced.

If these aspects are combined, the result is that, in an extreme flyingsituation, as shown in FIG. 3, the attitude correction with respect torolling and pitching angles should have priority over control concerningthe other control axes (yaw and thrust). According to FIG. 4, in thecase of the aerial vehicle shown with 18 rotors (cf. FIG. 1), thefollowing procedures or assignments may therefore be adopted:

The control matrix is broken down (dissociated)—as shown in FIG. 4—intotwo sub-matrices D₁, D₂. In this case, u_(p,1 ∈)

² corresponds to the rolling moment and the pitching moment.Consequently, D₁∈

^(2×k), k=1, . . . , 18, describes the mapping of the actuator commandsu onto the rolling and pitching moments. u_(p,2)∈

² corresponds to the yawing moment and the total thrust, and D₂∈

^(2×k), k=1, . . . , 18, correspondingly maps the actuator commands uonto the yawing moment and the thrust.

FIG. 5 shows a possible sequence of the method according to theinvention. It begins in step S1, wherein the pseudo-control commandu_(p) is calculated, as described above in detail. In step S2, thecontrol matrix D is dissociated on the basis of the controlprioritization, as described above by way of example. For this purpose,(sensor) measured values etc. from step S2′ may be used, as likewisedescribed. According to the measured values, the control prioritizationmay also be changed, as also described. In step S3, the calculation ofu₁=D₁ ^(#)u_(p,1) is performed to solve the primary task with the aid ofthe sub control matrix D₁, as specified. In step S4, the control commandu₂ ^(org) for the secondary task is calculated. For this purpose,(sensor) measured values etc. from step S4′ may be used, for examplemeasured temperature values, as likewise already described. Then, instep S5, the projection of the non-primary (secondary) task or of thecorresponding control command u₂ ^(org) into the null space of theprimary task takes place, so that, if u₂, u₂∈

^(k), D₁u₂=0 represents a control command for the actuators forperforming the non-primary task. This has also been described in detailfurther above and also applies correspondingly to the further,low-priority tasks.

Step S6 includes an inquiry as to whether still further subordinate(non-primary, for example tertiary) tasks are to be solved. If yes (y),steps S4 (or S4′) and S5 are correspondingly adapted (for example withu₃ ^(org) etc.) and repeated for the further task(s). As soon as theinquiry in step S6 is answered with no (n), the method jumps to afterstep S7, where the complete control command is determined: =u₁+u₂+ . . ., depending on the number of tasks solved (see above). In step S8, thiscomplete control command is used for controlling the actuators, and themethod ends with step S9.

1. A method for controlling an overdetermined system with multiplepower-restricted actuators, wherein the actuators perform at least oneprimary task and a number of non-primary tasks, the method comprising:a) determining a pseudo-control command u_(p)∈

^(p′) based on a physical model of a system, said pseudo-control commandrepresents the torques (L, M, N) and a total thrust force (F) acting onthe system, b) determining a control matrix D, D∈

^(p′×k) according to u_(p)=Du, wherein u denotes actuator controlcommands, c) dissociating the control matrix D into sub control matricesD_(i) according to ${D = \begin{bmatrix}D_{1} \\ \vdots \\D_{q}\end{bmatrix}},$ q≤p′, so that $u_{p} = \begin{bmatrix}u_{p,1} \\ \vdots \\u_{p,q}\end{bmatrix}$ and u_(p,i)=D_(i)u, wherein the sub control matricesD_(i) and corresponding sub pseudo-control commands u_(p,i) for i=1correspond to the primary task and for i>1 correspond to the at leastone non-primary task and wherein a priority of the non-primary tasksdecreases with increasing index i, d) determining actuator controlcommands for solving the primary task according to u₁=D₁ ^(#)u_(p,1),where # denotes a matrix inversion and u₁∈

^(k) comprises said actuator control commands which solve the primarytask, with u₁∈U:={u∈

^(k)|u^(min)≤u≤u^(max)} e) projecting the non-primary tasks, i>1, into anull space of the primary task, i=1, and into respective null space ofall of the non-primary tasks of higher priority, if present, so thatD_(i)u_(j)=0, with u_(j)∈

^(k), j>i, comprises actuator control commands for performing anon-primary task, and f) providing the actuator control commands from d)and e) at the actuators.
 2. The method as claimed in claim 1, furthercomprising first calculating u₂ ^(org)∈

^(k), in order to solve a first one of the non-primary tasks, accordingto u₂ ^(org)=D₂ ^(#)(u_(p,2)−D₂u₁), and subsequently projecting u₂^(org) onto the null space N₂(D₁) of the sub control matrix D₁ accordingto u₂=N₂u₂ ^(org), wherein N₂=I−D₁ ^(T)(D₁ ^(#))^(T).
 3. The method asclaimed in claim 2, further comprising subsequently calculating acomplete actuator control command u as follows:u=u ₁ +u ₂.
 4. The method as claimed in claim 3, wherein for u₂ ^(org)it is specified that u₂ ^(org)∈U₂ :={u ₂ ^(org)∈

^(k) |u ^(min) −u ₁ ≤N ₂ u ₂ ^(org) ≤u ^(max) −u ₁}.
 5. The method asclaimed in claim 2, further comprising subsequently calculating u₃^(org)∈

^(k) in order to solve a second one of the non-primary tasks, accordingto u₃ ^(org)=D₃ ^(#)(u_(p,3)−D₃(u₁−u₂)), and subsequently projecting u₃^(org) onto the null space N₃(D₁,D₂) of the sub control matrices D₁,D₂according to u₃=N₃u₂ ^(org), wherein$N_{3} = {\left( {I - {\begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{T}\left( \begin{bmatrix}D_{1} \\D_{2}\end{bmatrix}^{\#} \right)^{T}}} \right).}$
 6. The method as claimed inclaim 5, further comprising subsequently calculating a complete actuatorcontrol command u as follows: u=u₁+u₂+u₃.
 7. The method as claimed inclaim 6, wherein for u₃ ^(org) it is specified that u₃ ^(org)∈U₃:={u₃^(org)∈

^(k)|u^(min)−u₁−u₂≤N₃u₃ ^(org)≤u^(max)−u₁−u₂}.
 8. The method as claimedin claim 1, further comprising calculating the null space N_(i)(D), i>1according to N i ( D ) = ( I - [ D 1 ⋮ D i - 1 ] T ⁢ ( [ D 1 ⋮ D i - 1 ]# ) T ) ∈ k × k .
 9. The method as claimed in claim 8, furthercomprising for solving a non-primary task, i>1, calculating:u _(i) ^(org) =D _(i) ^(#)(u _(p,i) −D _(i)Σ_(j=1) ^(i−1) u _(j))u_(i)=N_(i)u_(i) ^(org);u=Σ _(i=1) ^(n) u _(i); withu _(i)∈U_(i) :={u _(i)∈

^(k) |u ^(min) −u _(Σ) ≤N _(i) u _(i) ≤u ^(max) −u _(Σ)} and withu _(Σ)=Σ_(j=1) ^(i−1) u _(j).
 10. The method as claimed in claim 1,wherein the method is applied to an aircraft (1) with multiplepropulsion units (3), and said propulsion units form at least some ofthe actuators of the system.
 11. The method as claimed in claim 1,further comprising prioritizing attitude control concerning rolling andpitching, over directional control concerning yawing and verticalcontrol concerning a flying altitude.
 12. The method as claimed in claim11, further comprising selecting D₁∈

^(2×k) such that it represents a mapping of the actuator controlcommands u onto a rolling and pitching moment, with u_(p,1)∈

², and selecting D₂∈

^(2×k) such that it represents a mapping of the actuator controlcommands u onto a thrust and yawing moment, with u_(p,2)∈

².
 13. The method as claimed in claim 10, further comprisingprioritizing attitude control concerning rolling and pitching overdirectional control concerning yawing and a total thrust.
 14. The methodas claimed in claim 13, further comprising selecting D₁∈

^(2×k) such that it represents a mapping of the actuator controlcommands u onto a rolling and pitching moment, with u_(p,1)∈

², selecting D₂∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto a yawing moment, with u_(p,2)∈

, and selecting D₃∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto the total thrust, with u_(p,3)∈

.
 15. The method as claimed in claim 10, further comprising prioritizinga total thrust over attitude control concerning rolling and pitching,and over directional control concerning yawing.
 16. The method asclaimed in claim 15, further comprising selecting D₁∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto the total thrust, with u_(p,1)∈

², selecting D₂∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto a pitching moment, with u_(p,2)∈

, selecting D₃∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto a rolling moment, with u_(p,3)∈

, and selecting D₄∈

^(1×k) such that it represents a mapping of the actuator controlcommands u onto a yawing moment, with u_(p,4)∈

.
 17. A control unit (2 a) for controlling an overdetermined system withmultiple power-restricted actuators, wherein the actuators perform atleast one primary task and at least one non-primary task, comprising acontroller configured to a) determining a pseudo-control command u_(p)∈

^(p′) based on a physical model of a system, said pseudo-control commandrepresents torques (L, M, N) and a total thrust force (F) acting on thesystem, b) determining a control matrix D, D∈

^(p′×k), according to u_(p)=Du, wherein u denotes actuator controlcommands, c) dissociating the control matrix into sub control matrices${D = \begin{bmatrix}D_{1} \\ \vdots \\D_{q}\end{bmatrix}},$ q≤p′, so that $u_{p} = \begin{bmatrix}u_{p,1} \\ \vdots \\u_{p,q}\end{bmatrix}$ and u_(p,i)=D_(i)u, wherein the sub control matricesD_(i) and corresponding sub pseudo-control commands u_(p,i) for i=1correspond to the primary task and for i>1 correspond to the at leastone non-primary task and wherein a priority of the non-primary tasksdecreases with increasing index i, d) determining the actuator controlcommands for solving the primary task according to u₁=D₁ ^(#)u_(p,1),where # denotes a matrix inversion and u₁∈

^(k) comprises said actuator control commands which solve the primarytask, with u₁∈U:={u∈

^(k)|u^(min)≤u≤u^(max)}, e) projecting the non-primary tasks, i>1, intoa null space of the primary task, i=1, and into respective null space ofall of the non-primary tasks of higher priority, if present, so thatD_(i)u_(j)=0, if u_(j)∈

^(k), j>i, comprises the actuator control commands for performing anon-primary task, and f) providing the actuator control commands from d)and e) at the actuators.
 18. The control unit (2 a) as claimed in claim17, wherein the controller is further configured such that it firstcalculates u₂ ^(org)∈

^(k), in order to solve a first one of the non-primary tasks, accordingto u₂ ^(org)=D₂ ^(#)(u_(p,2)−D₂u₁), and subsequently projects u₂ ^(org)onto the null space N₂(D₁) of the sub control matrix D₁ according tou₂=N₂u₂ ^(org), wherein N₂=I−D₁ ^(T)(D₁ ^(#))^(T).
 19. The control unit(2 a) as claimed in claim 17, further comprising at least one of devices(4) for at least one of measuring or determining parameters and statesof the system or the actuators which are required for the determinationof the pseudo-control command according to step a) connected to thecontroller.
 20. An overdetermined system comprising multiple actuatorsactively connected to the control unit (2 a) as claimed in claim
 17. 21.An aircraft (1) comprising the overdetermined system as claimed in claim20, wherein the actuators multiple propulsion units (3).